An induction motor, you drive flux into the stator, which induces a current in the rotor. The induced current in turn reflects a magnetic field, which serves as the permanent magnet in a PMAC machine.
The key observations are:
1. The reflected field is proportional to applied field.
The most direct consequence of this is, because flux is limited by stator cross section and flux density, the induced current is small at low frequencies. In other words, a VFD must reduce voltage in proportion to frequency, and the available torque (before stall) is proportional as well. So power goes as freq^2, and induction motors are not very useful at low RPM.
This does mean they are very useful above rated speed, given that core losses and ultimate RPM limits are respected, of course!
A 1kW size motor isn't going to last very long driven at say 400Hz+... but will be an impressive show when it grenades.

More modest frequencies (say 25-100Hz) are the useful variable range.
2. The field is not permanent. It is trapped in a conductor, but a nonideal one, so it slips.
After all, if you had a superconducting rotor, it would be a synchronous machine (although a slightly odd one where the reflected field is proportional to applied field). Or
possibly it wouldn't work because the induced current needs to lag behind the applied field, but I don't think so? (I'd have to refresh my knowledge of both motors and superconductors to verify that.)
Of course, if we charge the superconductor with a current (and thus field), it will be permanent, and will act exactly like a PM rotor. (Superconductors exclude external currents, so you can't quite do this in a single step, from the stator field. It would be an ordinary electromagnet rotor, synchronous machine, that happens to use a superconductor.)
Anyway, the rotor is synchronous with the stator's magnetic field as usual, but because the rotor's own field is not trapped -- it rotates within the rotor itself -- the total RPM is a bit less than the applied mains. This is why induction motors are labeled 1425 RPM or thereabouts (1750 RPM in US). The difference is called
slip, and is the rate the field rotates in the rotor. Yep, the eddy currents flowing in the rotor, rotate quite slowly indeed -- this corresponds to an L/R time constant (of the rotor) of a few Hz, not bad, eh?
You can also demonstrate this at ~0 RPM, by biasing the stator with some DC current, and turning the rotor. This is a highly effective magnetic brake, and this method of operation is sometimes used to stop machinery.
Indeed, the rate of slip is proportional to applied torque, so this could also be used to some extent to gauge load factor; it also means that an induction machine can be used perfectly symmetrically as a generator, you just need to push it faster than the synchronous speed.
You can use an induction machine as a generator (alternator), after supplying some seed current. Usually there's enough residual magnetism in the rotor steel to get things going, and a capacitor is used (I think) to maintain current between cycles. Starting a motor from no current, the generated voltage increases exponentially until limited by saturation (which will be close to nominal output voltage, because the stator is designed to run just shy of saturation), and then it's about as good as a synchronous machine; give or take the waveform, which may be ratty due to the saturation, and of course having poor frequency stability because of slip.
Tim